How to Calculate the Load Pole and ESR Zero When Using Hybrid Output Capacitors

Application Report SLVAE26A–September 2018–Revised April 2019

Hao Zhang, Jason Wang

ABSTRACT Multi-layer ceramic (MLCC), aluminum electrolytic, tantalum, and polymer are the capacitor types most widely used in DC/DC switching regulator circuits. In practical applications, a power designer can use a hybrid capacitor network, formed by combining different capacitor types, in an effort to achieve low ESR and high capacitance. This can be a very effective method of reducing output ripple and improving load transient performance. This application report provides a method to analyze how the hybrid capacitor network affects the loop. The Section 1 section introduces the key characteristics of each of the different types of capacitors. The Section 2 section discusses peak current mode power stage small signal modeling with a hybrid output capacitor network. Finally, the Section 3 section verifies the analysis with bench test results on TPS65400EVM.

Contents 1 Introduction ...... 2 2 Current Mode Power Stage Small Signal Modeling...... 3 3 Bench Verification ...... 7 4 Summary ...... 12 5 References ...... 12

List of Figures 1 Simplified Current Mode Functional Block Diagram ...... 3 2 Current Mode Buck Converter Circuit With Hybrid Output Capacitor Network ...... 4 3 Simplified Power Stage Bode Plot of Single Output Capacitor Configuration ...... 6 4 Simplified Power Stage Bode Plot of Hybrid Output Capacitor Network Configuration ...... 7 5 Simplified Schematic of All MLCC Output Capacitor Configuration...... 8

6 Bode Plot of All MLCC Output Capacitor Configuration (VOUT = 3.3 V, IOUT = 1.5 A)...... 8 7 Simplified Schematic of Hybrid Output Capacitor Configuration ...... 9 8 Bode Plot of Hybrid Output Capacitor Configuration ...... 10 9 Simplified Schematic of Single Polymer Output Capacitor Configuration ...... 10

10 Bode Plot of Single Polymer Output Capacitor Configuration (VOUT = 3.3 V, IOUT = 1.5 A) ...... 11 11 Gain Frequency Response Comparison Among Hybrid, MLCC, and Polymer Output Capacitor Configuration ...... 11

List of Tables 1 Comparison of Different Types of Capacitors...... 2 2 Basic Configurations of TPS65400EVM ...... 7 3 Parameters of Capacitors Used in Output Capacitor Network...... 9 Trademarks All trademarks are the property of their respective owners.

SLVAE26A–September 2018–Revised April 2019 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output 1 Submit Documentation Feedback Capacitors Copyright © 2018–2019, Texas Instruments Incorporated Introduction www.ti.com 1 Introduction Output capacitors are critical components for a switch-mode power supply. To select output capacitors for a DC/DC switching regulator application, the basic parameters you must pay attention to include capacitance, equivalent series resistance (ESR), rated voltage, and size. Table 1 compares the four types of capacitors.

Table 1. Comparison of Different Types of Capacitors

CAPACITANCE ESR RATED VOLTAGE SIZE Multi-layer Ceramic Low Low Medium Small Capacitor Aluminum Electrolytic High High Various Large Capacitor Tantalum Capacitor Medium Medium Medium Medium Polymer Capacitor High Medium Medium Medium

Generally speaking, MLCC provides very low ESR, which is critical to compress the resistive output ripple. Aluminum electrolytic capacitors provide a large amount of capacitance, but have the highest ESR among the four capacitor types. Tantalum and polymer capacitors have medium-range capacitance values, ESR, and rated voltage. By using a hybrid capacitor network, designers can take advantage of the benefits of each capacitor type. In applications where small ripple, overshoot, and undershoot are required, hybrid output capacitor networks are very common. Loop stability is another important topic for DC/DC switching regulator circuit design. In DC/DC converter small signal modeling, the capacitance and ESR values of the output capacitor have a direct effect on the poles and zeros in the open loop transfer function. With the presence of a hybrid output capacitor network, new poles and zeros are introduced into the loop by the network itself. This application report discusses how hybrid output capacitors influence the loop, and then verifies the analysis using the TPS65400EVM.

2 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output SLVAE26A–September 2018–Revised April 2019 Capacitors Submit Documentation Feedback Copyright © 2018–2019, Texas Instruments Incorporated www.ti.com Current Mode Power Stage Small Signal Modeling 2 Current Mode Power Stage Small Signal Modeling Figure 1 shows the simplified functional block diagram of a peak current mode DC/DC circuit.

Current + Sense

œ VIN Power stage

dÙ

vÙ Control out L Logic ESR Rload RFBT COUT

vÙFB

RFBB œ

vÙ COMP Compensation EA + Network VREF Control stage

Figure 1. Simplified Current Mode Functional Block Diagram

To analyze and judge the loop stability by open loop transfer function and Bode plot, the loop is split into two components:

Gopen )s( = Gdv ∂H)s( vd )s(

where

• Gdv(s) are the transfer function from control to the output ( to ), including the PWM modulator and power stage.

Ù Ù • Hvd(s) is the transfer function from the output to control ( vout to d ), including the feedback path, compensation network. (1) Poles and zeros in power stage have a direct effect on the loop transfer function. To understand how the hybrid output capacitor network affects the loop, calculate the poles and zeros in the power stage. The calculation can vary in the different control mode, as they have different control to output transfer function

Gdv(s). This application report shows how to do the calculation based on a current mode DC/DC converter circuit. Impedance of hybrid output capacitor network is discussed first. Based on that, the effect of the network on current mode DC/DC circuit is analyzed.

SLVAE26A–September 2018–Revised April 2019 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output 3 Submit Documentation Feedback Capacitors Copyright © 2018–2019, Texas Instruments Incorporated Current Mode Power Stage Small Signal Modeling www.ti.com 2.1 Impedance of Hybrid Output Capacitor Network Figure 1 shows that the output capacitor in the functional block diagram is simplified as a single capacitor. Impedance of a single capacitor with a specific ESR is: 1+ s ∂ESR ∂ C Zcap )s( = sC (2) When the output capacitor is substituted by an output capacitor network, the situation can be different.

Figure 2 shows two capacitors with same or different capacitance and ESR that are put in parallel.

+ Current œ Sense VIN

Hybrid Output Capacitor Netwrok Vout Control Logic L ESR1 ESR2 RFBT Rload

Cout1 Cout2

RFBB

GND œ

Compensation EA + Network VREF

Figure 2. Current Mode Buck Converter Circuit With Hybrid Output Capacitor Network

Impedance of the hybrid capacitor network Zcap(s) can be expressed as: 1 1 Zcap )s( = R( 1 + R//() 2 + ) sC1 sC 2

where

• R1 is the ESR of C1

• R2 is the ESR of C2 (3) If you expand Equation 3, you have: 1( + sR ∂ 1()C + sR C ) Z )s( = 11 2 2 cap » ÿ C C 21 C(s 1 + C 2 ) ∂ …1+ ∂ R(s 1 + R 2 ) ∂ Ÿ C + C ⁄ 1 2 (4) Equation 4 indicates when the capacitance and ESR are different. There is an initial pole, a parallel pole, and two ESR zeros in the impedance expression. A specific situation is where the two capacitors are same, C1 = C2 = C, R1 = R2 = R. Equation 4 can be further simplified as:

4 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output SLVAE26A–September 2018–Revised April 2019 Capacitors Submit Documentation Feedback Copyright © 2018–2019, Texas Instruments Incorporated www.ti.com Current Mode Power Stage Small Signal Modeling

1+ sRC Zcap )s( = ∂ C2s (5) In this situation, two of the same output capacitors can be equalized to one single capacitor with two times capacitance and half ESR.

2.2 Pole and Zero Calculation in Current Mode Power Stage The distribution of poles and zeros in the loop transfer function depends on both power stage characteristic and control mode. As discussed in the Current-Mode Modeling for Peak, Valley and Emulated Control Methods Application Report, the transfer function of peak current mode buck converter power stage is: s 1 + Ù w v out = 1 ∂ ESR Ù R 1 s s v c i + 1( + ∂ 1() + ) w w R o K m 0 L

where

• Ro is the loading resistance

VIN K = m V • Km is the modulator voltage gain, which is given by ramp

• Ri is the product of current sense gain and current sense resistance

• ω0 is the dominant pole

• ωL is the inductor pole

• ωESR is the ESR zero of the output capacitor (6) The dominant pole locates at the frequency where the impedance of output capacitor equals to the loading resistance:

(7) The inductor pole locates at the frequency where the impedance of the inductor equals to the current-loop gain: ∂ w = Km Ri L L (8)

In most cases, Km × Ri » 1, so ω0 can be simplified as:

w = 1 0 C ∂R out o (9)

Also, as Km × Ri » 1, ωL is usually much higher than the bandwidth, which means the pole has limited effect on the loop stability and can be ignored in the following analysis. The ESR zero of output capacitor locates at: 1 w = ESR ì ESR COUT (10)

SLVAE26A–September 2018–Revised April 2019 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output 5 Submit Documentation Feedback Capacitors Copyright © 2018–2019, Texas Instruments Incorporated Current Mode Power Stage Small Signal Modeling www.ti.com Equation 6 is further simplified as: s 1 + Ù w v out = R o ∂ ESR Ù s v c R i 1 + w 0 (11) Figure 3 shows the simplified Bode plot.

Gain

Loading pole w 0

ESR zero w ESR

Frequency

Figure 3. Simplified Power Stage Bode Plot of Single Output Capacitor Configuration

When the output capacitor is substituted by a hybrid capacitor network, modify Equation 11 as: s s 1( + 1)( + ) Ù w w v out = R o ∂ ESR 1 ESR 2 vÙ R s s c i 1( + 1)( + ) w w 0 p

where

w = 1 0 ∂ + • ω0 is the dominant loading pole in current mode, given by Ro C( 1 2)C

w = 1 p C C R( + R ) ∂ 21 1 2 + • ωp is the parallel pole introduced by the parallel capacitors, given by C1 C2

• ωESR1 and ωESR2 are the ESR zeros of the capacitors, given by

w = 1 w = 1 ESR1 ∂ ESR2 ∂ ESR1 C1 ESR2 C2 (12)

6 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output SLVAE26A–September 2018–Revised April 2019 Capacitors Submit Documentation Feedback Copyright © 2018–2019, Texas Instruments Incorporated www.ti.com Bench Verification Figure 4 shows the simplified Bode plot.

Gain

Loading pole w0

ESR zero w ESR1

w p Parallel pole

ESRw zero ESR2

Frequency

Figure 4. Simplified Power Stage Bode Plot of Hybrid Output Capacitor Network Configuration

If you compare Equation 11 and Equation 12, you have:

● The dominant pole ω0 is determined by loading resistance and the sum of all capacitance in the power stage. ● ESR zero of the electrolytic capacitor remains at the same frequency.

● An additional pole ωp is introduced by the two different capacitors in output capacitor network. Take these poles and zeros into consideration when designing the loop compensation with a hybrid output capacitor network. These results are verified on a TPS65400EVM in Section 3.

3 Bench Verification

3.1 TPS65400 Introduction The TPS65400 is a 4.5-V to 18-V, synchronous quad buck converter with PMBus/I2C interface. The device works in peak current mode with an external loop compensation network. To support high output

capacitance in this test, compensation zero formed by Rcomp and Ccomp is placed at 0.6 kHz. The rolling pole formed by Rcomp and Croll locates at 141.6 kHz. The test is performed on the first channel of the TPS65400EVM. Table 2 lists some of the basic configurations. See the TPS65400 EVM User's Guide for other detailed descriptions and configurations.

Table 2. Basic Configurations of TPS65400EVM

VIN 12 V VOUT 3.3 V Switching Frequency 500 kHz Loading Current 1.5 A Loading Pole 0.33 kHz Compensation Zero 0.6 kHz Rolling pole 141.6 kHz

3.2 Bench Test To verify the calculation in the previous section, a Bode plot is done for the output capacitor network formed by all MLCC, hybrid, and single polymer capacitors.

3.2.1 All MLCC Figure 5 shows the output capacitor network is formed all by MLCC. The part number of the capacitor is GRM31CR61A476KE15L. Taking the DC bias effect into consideration, the total effective capacitance is about 224.8 µF. Figure 6 shows the Bode plot of all MLCC configuration as the output capacitor.

SLVAE26A–September 2018–Revised April 2019 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output 7 Submit Documentation Feedback Capacitors Copyright © 2018–2019, Texas Instruments Incorporated Bench Verification www.ti.com

All MLCC 12V Vout=3.3V Iout=1.5A VIN SW L 2.2uH

RFBT TPS65400 Rload Cout1 Cout2

47uF*8 COMP FB RFBB Rcomp 51.1k GND Croll 22pF Ccomp 4.7nF

Figure 5. Simplified Schematic of All MLCC Output Capacitor Configuration

Figure 6. Bode Plot of All MLCC Output Capacitor Configuration (VOUT = 3.3 V, IOUT = 1.5 A)

According to the characteristic curve, the capacitor ESR is about 3 mΩ. ESR zero of single capacitor locates at about 1.95 MHz, which is much higher than the bandwidth. As in the previous analysis, putting the capacitors in parallel has no effect on the frequency of ESR zero. That is to say, besides the error amplifier pole, the loading pole and the compensation zero, there is no any other pole nor zero within the bandwidth. The gain curve keeps going down with a –20 dB/dec slope after the loading pole and compensation zero. The bandwidth is about 22.2 kHz, and the phase margin is 62.9 degrees.

8 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output SLVAE26A–September 2018–Revised April 2019 Capacitors Submit Documentation Feedback Copyright © 2018–2019, Texas Instruments Incorporated www.ti.com Bench Verification 3.2.2 Hybrid Output Capacitor Network Figure 7 shows the hybrid output capacitor network is formed by one polymer capacitor and four MLCCs. Total effective output capacitance is about 222 µF. Table 3 lists the parameters of the components.

Hybrid Output 12V Capacitor Vout=3.3V Iout=1.5A VIN SW L 2.2uH 150uF/ 22uF*4 50mΩ RFBT TPS65400 Rload Cout1 Cout4 Cout5

COMP FB RFBB Rcomp 51.1k GND Croll 22pF Ccomp 4.7nF

Figure 7. Simplified Schematic of Hybrid Output Capacitor Configuration

Table 3. Parameters of Capacitors Used in Output Capacitor Network

PART NUMBER CAPACITANCE/µF ESR /mΩ COUNT 16TQC150MYF 150 50 1 GRM31CR71A226KE 22 3 4 15L

To analyze this hybrid network, the same four MLCC can be equalized to one single MLCC by Equation 5. Then, according to the Equation 12, in the hybrid output capacitor network, ESR zero of the polymer capacitor locates at Equation 13. 1 f = = 2.21 kHz ESR1 2 ∂p ESR ∂ C 1 1 (13) The pole formed by putting different capacitors in parallel locates at Equation 14. 1 f = = 8.61 kHz p 2 ∂p (ESR + ESR ∂ C() C// ) 1 2 1 2 (14)

Figure 8 shows the Bode plot. The existence of the polymer capacitor ESR zero fESR1 and the parallel pole fp extends the bandwidth to 43 kHz, with a phase margin of 79.7 degrees. Judging from the gain curve, the zero and pole can be identified at the calculated frequency above. At a higher frequency, the parallel pole

together with the rolling pole formed by Rcomp and Croll keeps the gain curve rapidly going down.

SLVAE26A–September 2018–Revised April 2019 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output 9 Submit Documentation Feedback Capacitors Copyright © 2018–2019, Texas Instruments Incorporated Bench Verification www.ti.com

Figure 8. Bode Plot of Hybrid Output Capacitor Configuration

3.2.3 Single Polymer Capacitor Figure 9 shows the output capacitance consists of only one polymer capacitor with high ESR value.

Single Polymer 12V Output Capacitor Vout=3.3V Iout=1.5A VIN SW L 2.2uH

RFBT TPS65400 Rload Cout 220uF/ 35mΩ

COMP FB RFBB Rcomp 51.1k GND Croll 22pF Ccomp 4.7nF

Figure 9. Simplified Schematic of Single Polymer Output Capacitor Configuration

The part number of the polymer capacitor is 6TPE220MAZB. It has a capacitance of 220 µF, with 35 mΩ ESR. ESR zero locates at Equation 15. 1 f = = 7.20 kHz ESR ∂p ∂ 2 ESR C (15)

10 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output SLVAE26A–September 2018–Revised April 2019 Capacitors Submit Documentation Feedback Copyright © 2018–2019, Texas Instruments Incorporated www.ti.com Bench Verification To make the comparison more convincing, the total effective output capacitance is set to be almost the same with the previous test. Figure 10 shows the Bode plot.

Figure 10. Bode Plot of Single Polymer Output Capacitor Configuration (VOUT = 3.3 V, IOUT = 1.5 A)

Because of the low frequency ESR zero, the gain curve is going down very slowly until the rolling pole (141 kHz) takes effect. The bandwidth is extended to 164 kHz, which has exceeded 1/5 switching frequency. Phase margin is about 35.6 degree. Obviously, the compensation network designed for other configurations is not doing well in this situation.

3.2.4 Comparison To show how the hybrid configuration affect the loop, the Bode plot of three configurations are drawn together. Figure 11 shows the gain-frequency response curves.

60 Hybrid MLCC 40 Polymer

Gain (dB) -20

-40

-60

-80 102 103 104 105 106 Frequency (Hz) Plot Figure 11. Gain Frequency Response Comparison Among Hybrid, MLCC, and Polymer Output Capacitor Configuration

Effective output capacitance is almost the same among the three configurations. However, MLCC configuration has the smallest bandwidth because there is no ESR zero introduced by the high ESR capacitor. For the hybrid configuration consisting of MLCC and polymer capacitors, the ESR zero locates at the same frequency with the standalone polymer capacitor configuration. As calculated in Equation 14, the parallel pole introduced by hybrid configuration locates at 61.8 kHz. It is the parallel pole which leads to the significant difference between the hybrid configuration and the standalone polymer capacitor.

SLVAE26A–September 2018–Revised April 2019 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output 11 Submit Documentation Feedback Capacitors Copyright © 2018–2019, Texas Instruments Incorporated Bench Verification www.ti.com 3.3 Conclusion From the analysis and bench verification, three conclusions can be drawn. • The frequency of loading pole depends on the total effective capacitance of the output capacitor network. • In the parallel output capacitor network, ESR zero of each capacitor is still taking effect independently. Also, by putting two output capacitors in parallel, a parallel pole is introduced into the loop. • The effect brought by the ESR zero and parallel pole depends on how close they are on the frequency axis. In most cases, due to the DC bias degrading of MLCC, solution size, and cost, the total effective capacitance of MLCC is usually much smaller than that of electrolytic or other capacitor with higher

w = 1 p + ∂ capacitance and ESR. See the expression of ωp, R( 1 2 C)R 1 C// 2 . It can be identified that the bigger difference between C1 and C2, the further the parallel pole and the ESR zero are, and the less cancellation between ESR zero and parallel pole. This means its Bode plot curve looks more like single output capacitor with high ESR. Take this into consideration when doing the loop compensation design.

4 Summary Hybrid output capacitor network with high capacitance is widely used to support applications with strict limitation on ripple, overshoot and undershoot. This application report analyzed the poles and zeros in the power stage when using a hybrid output capacitor network theoretically. Furthermore, the analysis is verified on TPS65400EVM. The conclusion can be a guidance for the compensation design of DC/DC circuit with hybrid output capacitor network.

5 References • Texas Instruments, TPS65400 4.5- to 18-V Input Flexible Power Management Unit with PMBus/I2C Interface Data Sheet • Texas Instruments, TPS65400EVM 4.5- to 18-V Input Flexible Power Management Unit with PMBus/I2C Interface Evaluation Module User's Guide • Texas Instruments, Current-Mode Modeling for Peak, Valley and Emulated Control Methods Application Report

12 How to Calculate the Load Pole and ESR Zero When Using Hybrid Output SLVAE26A–September 2018–Revised April 2019 Capacitors Submit Documentation Feedback Copyright © 2018–2019, Texas Instruments Incorporated www.ti.com Revision History

Revision History NOTE: Page numbers for previous revisions may differ from page numbers in the current version.

Changes from Original (September 2018) to A Revision ...... Page

• Edited the application report for clarity...... 1

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